Multi-criteria decision-making approach for multi- stage optimal ...

www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 17th August 2009 Revised on 25th July 2010 doi: 10.1049/iet-gtd.2009.0709

ISSN 1751-8687

Multi-criteria decision-making approach for multistage optimal placement of phasor measurement units R. Sodhi S.C. Srivastava S.N. Singh Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India E-mail: [email protected]

Abstract: An important step in the development of synchrophasor-based wide area monitoring and control system is to optimally place the phasor measurement units (PMUs) in the network. Optimal number of PMUs, to fully observe the power system network, may be large. Owing to their relatively high cost, utilities may like to install these devices in stages. A technique is proposed here for placing the PMUs in multiple stages over a given time period that ensures complete power system observability even under a branch outage or a PMU failure. The proposed scheme is based on integer linear programming (ILP) and a multi-criteria decision-making (MCDM) approach. The ILP is used to obtain the optimal PMU locations in the system to completely observe the system and, subsequently, the MCDM model is developed to prioritise these PMU locations. Three indices are proposed to develop the MCDM model, viz. bus observability index, voltage control area observability index and tie-line oscillation observability index. Finally, the PMU placement is carried out in stages based on the ranking of these optimal locations. The proposed method is applied on three test systems – IEEE 14-bus system, New England 39-bus system and Northern Regional Power Grid 246-bus Indian system.

1

Introduction

Synchrophasor-based wide area monitoring and control systems (WAMCS) are being popularly deployed by electric power utilities [1]. The major advantages of the WAMCS include dynamic wide area measurements with higher accuracy and faster rate. The time synchronised voltage and current phasors are estimated through phasor measurement units (PMUs) located in the field, facilitating systems wide monitoring and control. For utilising the synchrophasor technology in an optimal manner, the first and the foremost task is to determine a set of suitable locations in the system where the PMUs can be installed, incurring minimum cost while making the system observable. This problem has been termed as the optimal PMU placement (OPP) problem in the literature. The OPP problem was first formulated by Baldwin et al. [2], using graph – theoretic approach in order to ensure the system topological observability. The method utilised depth first search algorithm to find an initial set for PMU placement, which was further optimised by using simulated annealing technique. Miloˇsevic´ and Begovic [3] used a genetic algorithm-based method to place a minimal set of PMUs, considering two conflicting objectives, viz. minimisation of number of PMUs and maximisation of the measurement redundancy. An integer linear programming (ILP)-based formulation was suggested by Xu and Abur [4] for placing PMUs in the network to make it fully observable. Rakpenthai et al. [5] used the condition number of the normalised measurement matrix as a criterion for selecting candidate solutions, along with binary integer IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181– 190 doi: 10.1049/iet-gtd.2009.0709

programming to find the PMU locations. Chakrabarti and Kyriakides [6] suggested a binary search method to find out minimum number of PMUs to ensure complete observability of the power system. PMU placement has also been carried out for some other specific applications, such as voltage stability [7], transient stability [8] and fault location in the network [9]. However, a literature survey reveals that most of the OPP work has been carried out for making the power system fully observable with their minimal number. One apparent reason for making the system completely observability with only PMU measurements is that such a PMU placement will lead to the use of linear state estimator, which is much faster and accurate as compared to the conventional supervisory control and data acquisition-based non-linear state estimators [10]. Most of the available OPP techniques, in the literature, concentrate only on finding the optimal number and locations of the PMUs in a power system network. However, in a large practical system, the number of PMUs, to fully observe the system, may be very large. The high capital cost associated with the PMU installation corroborates the need of installing the PMUs in multiple stages. To the best of authors’ knowledge, the multi-stage PMU placement has been addressed by only Nuqui and Phadke [11] and Dua et al. [12] so far. The concept of depth of unobservability has been used in [11] to phase out the PMU placement. This method deals with the complete enumeration of spanning tress of the power system network graph, whereas an ILP-based algorithm has been devised for optimal multi-stage scheduling of PMUs in [12]. The 181

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www.ietdl.org objective of the OPP method, suggested by Dua et al. [12], is to achieve complete observability of the system over a time span, and therefore the overall PMU placement result, after the completion of the last stage of the PMU placement, comes out to be identical to that obtained by placing the PMUs together in a single stage using the ILP method, ensuring the full-system observability. In this work, a methodology is suggested first for determining the optimal number and locations of PMUs ensuring complete observability of the system and, subsequently, placing these PMUs in a phased manner utilising a multi-criteria approach, based on certain criteria like tie-line oscillation observability and voltage control area observability. The main contribution of this work is the development of the multi-criteria framework to prioritise different optimal PMU locations and, subsequently, using it to install PMUs in stages according to their relative ranking. In the first step of the proposed method, an ILP-based PMU placement method is carried out with the objective of finding minimum number of PMUs, required for rendering the complete system observability. The PMU placement considers contingency cases pertaining to single PMU outage and single branch outage. Subsequently, a multicriteria decision-making (MCDM) [13] model is utilised for selecting a subset of PMUs to be installed in stages, out of the total optimal PMUs set obtained through the ILP. The methodology is tested on three test systems, viz. IEEE 14bus system, New England (NE) 39-bus system and 246-bus Northern Region Power Grid (NRPG) Indian system to demonstrate its effectiveness.

2

Identifying optimal PMU locations

A minimal set of PMU locations can be obtained, for a given network topology, to ensure system observability. However, loss of a transmission line or a PMU may result into unobservability of some of the buses in the power system network. To determine a robust metering scheme and, thereby, to enhance the reliability of the system monitoring, each node should be observed through at least two PMUs. This constraint can be incorporated in the basic ILP-based PMU placement method suggested in [4] as follows. To ensure the system’s complete observability with minimum number of PMUs, following rules and assumptions have been applied in this work: 1. It is assumed that the PMUs provide two types of measurements, viz. the voltage phasors and the outgoing line current phasor measurements. 2. If voltage phasor and current phasor at one end of a branch are known, voltage phasor at the other end of the branch can be calculated using Ohm’s law. 3. Each substation is represented by a single ‘node/bus’. However, a substation might have a double busbar arrangement involving double or one and a half breaker scheme. It is assumed that a single PMU will be used to monitor voltage at the two busbars and also to monitor currents in all the outgoing feeders emanating from the substation. 4. The network parameters, that is, R, L, X, C and the system topology are assumed to be known. 5. In this study, cost of all the PMUs is assumed to be same. It is recognised that each of the PMUs may have different number of channels and their costs may vary accordingly. The proposed method, however, is general and can be used to accommodate different cost of the PMUs. 182 & The Institution of Engineering and Technology 2011

With the above-stated assumptions, the following notations have been used in this work: 2.1

Parameters

N – number of buses in the system ci – cost of installing a PMU at bus i aij – ijth entry of a connectivity matrix A defined as  1, if i and j are directly connected aij = 0, otherwise 2.2

Decision variable

ui ¼ 1 if PMU is placed over bus i, 0 otherwise U – vector containing the binary decision variables, ui The ILP problem can be formulated as N 

ci ui

(1)

subject to AU ≥ 2

(2)

ui = (0/1) ∀i

(3)

minimise

i=1

The objective function (1) is to minimise the total cost of PMU installation. In this study, cost of all the PMUs is assumed to be same. However, the proposed method can be used to accommodate different PMU costs by changing the value of ci in (1). Constraints (2) ensure that each bus in the network is observed by at least two PMUs. For the purpose of illustration, the above ILP formulation can be explained with the help of a simple 5-bus system as shown in Fig. 1. In the formulation, U contains five binary decision variables. The constraint at each bus is formulated to ensure that each bus is connected to two PMUs. To illustrate the constraints, consider the bus-2, connected to buses 1, 3 and 5. The bus-2 will be made observable by at least two PMUs if f2 : u2 + u1 + u3 + u5 ≥ 2

(4)

The above formulation of the PMU placement differs from [4] in the sense that right-hand side of the constraint (2) is two instead of one. This will ensure that each bus is observed by a minimum of two PMUs and will ascertain that a PMU outage or failure of a communication link does not lead to loss of observability. In [4], each bus is observed by at least one PMU, which may result in unobservability of some of the buses in case of a PMU or transmission line outage, as illustrated further.

Fig. 1 Network diagram of a 5-bus sample system IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181 –190 doi: 10.1049/iet-gtd.2009.0709

www.ietdl.org Assuming cost of PMU installation at each node as 1 p.u., the ILP algorithm for the sample system of Fig. 1 can be formulated as

minimise

5  j=1

s.t.

fj : uj +

uj = (0/1);

1. Identify various criteria to be used in the evaluation process and determine criteria value for each alternative. 2. Determine weights for different criteria. 3. Calculate the utility value for each alternative using the following equation

uj 

uIj ≥ 2;

criteria under consideration. The MAUT approach can be summarised in the following steps:

j = 1, 2, . . . , 5

j = 1, 2, . . . , 5

UVi =

k 

i = 1, 2, . . . , P

wj vij ;

(7)

j=1

where Ij is the set of all the buses directly connected to the bus j. The solution of the above ILP, using CPLEX software [14], is obtained as U∗ ¼ {2, 4, 5}. The solution of the ILP problem, using [4] is U∗ ¼ {2, 5}. It is clear, from Fig. 1, that using the optimal PMU placement solution as U∗ ¼ {2, 5}, bus-4 becomes unobservable in the event of failure of PMU at bus-5 or line 4 – 5 outage, whereas PMU placement solution U∗ ¼ {2, 4, 5} results in complete system observability even under such contingency cases. It must be noted that if a bus is observed by at least two PMUs, then a line outage will not affect the complete observability of the network. Thus, considering a PMU outage in optimal PMU placement problem also takes care of N 2 1 contingency cases.

3 Proposed multi-stage PMU placement approach Let the ILP algorithm (in Section 2) results in total P number of optimal PMU locations, which are to be placed in S stages. Let the number of PMUs to be placed in stage-i be ni such that SSi¼1ni ¼ P. So, the total number of available alternatives, L, for selecting a subset from optimal PMU locations, P, in the first stage will be L1 =

P! n1 ! × (P − n1 )!

(6)

The number of alternatives available for PMU placement in successive stages keep on reducing and the total number of available options can be numerous in case of a large power system. In order to obtain the best alternative, out of a large number of available options, the problem can involve a number of objectives, which have to be addressed simultaneously. Thus, the multi-stage optimal PMU installation becomes a MCDM problem. The decision of selecting a suitable subset of optimal PMU locations, in each stage, can involve addressing many important factors such as observing the tie-line oscillations, voltage control areas, critical corridors and so on. An MCDM model is adopted for this purpose as described below. Several methods are available in the literature for MCDM. A multi-attribute utility theory (MAUT) [15] approach has been used in this study as it is quite simple and an intuitive approach to the decision making. Additionally, it allows the decision maker to allocate relative weights to various IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181– 190 doi: 10.1049/iet-gtd.2009.0709

3.1

PMU selection criteria

In the present work, superiority of a particular PMU location has been judged based upon three criteria. These include bus voltage observability, voltage control area observability and tie-line oscillations observability. To incorporate these three criteria in the MCDM model, three respective indices have been defined as given below. 3.1.1 Bus voltage observability index (BOI): BOI for a PMU bus p is defined as the number of buses directly connected to the PMU bus. The BOI at a PMU bus p, in an N-bus system, is, therefore, expressed as BOIp =

N 

Cpj

(8)

j=1 j=p

(5)

Similarly, available alternatives for placing the n2 PMUs in the second stage becomes (P − n1 )! L2 = n2 ! × (P − n1 − n2 )!

where UVi is the utility value of alternative i, wj represents the weight assigned to criterion j, vij is the value associated with alternative i for criterion j, k is the total number of criteria and P is the total number of alternatives available. Various steps of MAUT, used in this study for multi-stage PMU placement, are given in the following subsection.

 where

Cpj =

1, 0,

if p and j are directly connected otherwise

To enhance the observability of the power system with phasor measurements, a bus having maximum connectivity with other buses becomes a preferred choice for the PMU placement. 3.1.2 Voltage control area observability index (VOI): In initial stages, PMUs should be uniformly distributed in the system. This can be ensured by selecting the PMU locations that are electrically far apart. This would also assist in identifying different voltage control areas in the power system [16] along with uniform distribution of PMUs in each stage. Electrical distance between two nodes i and j is calculated as follows: 1. Calculate Newton– Raphson load flow Jacobian J and, hence, obtain the sub-matrix J4 , where J4 ¼ [∂Q/∂V ]. 2. Invert J4 . Say B ¼ [∂Q/∂V ]21 ¼ J21 4 . The elements of B are written as bij ¼ (∂Vi/∂Qj). 3. Obtain attenuation matrix aij , between nodes i and j, as aij ¼ bij/bjj . 4. Calculate the electrical distance between nodes i and j as Dij ¼ Dji ¼ 2log10(aij × aji). 183

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www.ietdl.org Once electrical distances between every pair of nodes are calculated, voltage control area observability index (VOI) for a PMU bus-p can be defined as VOIp =



Dpj ,

p [ PMU buses

(9)

j[ PMU buses

Dpp ¼ 0, as electrical distance at the same bus will be zero. 3.1.3 Tie-line oscillation observability index (TOI): It is desirable to monitor the tie-line buses, because the power oscillations in large systems are generally observed through the dynamic changes in the tie-line power flows and phase angle difference at its two ends [10]. A good PMU placement strategy should try to make as many tie-lines observable as possible in every stage of the PMU placement. Since PMUs, as given in Section 2, are placed in such a way that each bus is observed by at least two PMUs, six different cases might arise as depicted in Figs. 2a – e. The heuristic approach, used to calculate TOI, in each of these cases is discussed below. 1. Fig. 2a depicts the case when both the tie-line buses m and n are in the list of optimal PMU locations (as obtained in

Section 2). Since both PMUs are to be placed at tie-line buses over a period of time, both these locations are of equal importance and can be assigned with the same value of the tie-line index, say TOIm ¼ TOIn ¼ 1. 2. In Fig. 2b, a PMU is to be placed at tie-line bus m, whereas tie-line bus n can be observed by either of the PMUs, placed at bus m and bus l. Hence, it becomes important to place PMU at bus m to observe the tie-line n at least from one end during the initial stages. Therefore TOIm ¼ 1.5 and TOIl ¼ 0.5. 3. Fig. 2c shows a case when both the tie-line buses m and n are not in the list of optimal PMU placement and will be observed through PMUs to be placed at buses k and l. Since the tie-line m 2 n is going to be observed only via neighbouring PMUs to be placed at buses k and l, an early observability of the tieline can be assured by assigning highest index value to the respective PMU buses as TOIk ¼ TOIl ¼ 2.0. 4. Fig. 2d depicts a possible situation, where both the tie-line buses m and n are equipped with PMUs, and there is another tie-line m ′ 2 n ′ , in which bus m ′ is equipped with a PMU and its other corresponding tie-line bus n ′ is not in the list of optimal PMU locations. The network topology is such that bus n is a neighbouring bus of bus-n ′ . It is basically a simultaneous occurrence of case (1) at tie-line m 2 n and of case (2) at tie-line m ′ 2 n ′ . In such cases, TOIm ¼ 1, TOIm ′ ¼ 1.5, TOIn ¼ 1 + 0.5 ¼ 1.5. 5. Another possible case is shown in Fig. 2e, where bus l is a common neighbouring bus of two tie-line buses, viz. n and n ′ , equipped with a PMU. In such cases, TOI can be calculated as TOIm ¼ TOIn ¼ 1.5, TOIl ¼ 0.5 + 0.5 ¼ 1. 6. If a PMU bus location i (in the optimal PMU list) is not a tie-line bus, TOIi ¼ 0. 3.2

Weight calculation

Selection of weights, associated with each criterion, involves a certain amount of subjectivity and this feature can be very useful in the problem at hand. This is because there might be some critical/pressing factors, for example, tie-line observability, which need to be addressed with the top most priority in a multi-staged PMU placement. Such crucial criteria can always be assigned a highest weight in the proposed MCDM method and, thereby, an early PMU placement can be ensured to handle such bottlenecks. For calculating the weights of different criteria, the principle of analytic hierarchy process (AHP) [17] is used in this work. A pairwise matrix is constructed using AHP, which decides the relative importance of different criteria. The steps of the AHP algorithm can be briefly summarised as follows: 1. Form a pairwise matrix. The values of elements in this matrix reflect the user’s knowledge about relative importance between every pair of criteria. 2. Calculate the maximum eigenvalue and the eigenvector of the pairwise matrix. 3. Perform the hierarchy ranking. Thus, according to the above algorithm, pairwise matrix for the three criteria is formed as Fig. 2 Different cases for the calculation of tie-line observability index a b c d e

Both tie-line buses are equipped with PMUs One tie-line bus is equipped with PMU None of the tie-line buses is equipped with PMU One tie-line bus is neighbour of another tie-line bus Two tie-line buses have a common neighbouring bus

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⎡

w1 /w1 M = ⎣ w2 /w1 w3 /w1

w1 /w2 w2 /w2 w3 /w2

⎤ w1 /w3 w2 /w3 ⎦ w3 /w3

(10)

where wj is the weighting coefficient of criterion j, required to be determined. However, wj/wk , that is ratio of the weights for IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181 –190 doi: 10.1049/iet-gtd.2009.0709

www.ietdl.org all pairs of criteria, is assumed according to their relative importance [13]. The importance of one factor over another can be the integer numbers on a scale of 1 – 9 as follows: † † † † †

1 ¼ Equally important; 3 ¼ Moderately important over another; 5 ¼ Essentially important; 7 ¼ Very strongly important; 9 ¼ Extremely important.

The following equation is used to obtain the weights denoted by wj , in (7), as Mw = lmax w

(11)

where w is the three-dimensional eigenvector associated with the largest eigenvalue of pairwise matrix M. Elements of w are normalised to sum to unity. 3.3

Calculating utility value

With the availability of values of the proposed indices (calculated as described in Section 3.1) and the weights of the three criteria (Section 3.2), the utility value of each alternative can be calculated. The utility value is termed as PMU location performance index (PLPI) and is calculated as follows PLPIi =

3 

wj vij ;

i = 1, 2, . . . , P

Fig. 3 Flow diagram of the proposed multi-criteria multi-stage optimal PMU placement

(12)

j=1

where PLPI1 , PLPI2 , . . . , PLPIP are the final ranking values of each PMU location. 3.4

Normalisation of criteria and weights

Normalisation process facilitates the relative comparison between values of different criteria. The value of each criterion, calculated from (8) and (9), has been normalised to range from zero to one. In general, if c is a vector with its ith value as ci , then the vector c can be normalised in the range g, g + k as cˆ i =

k(ci − cmin ) +g (cmax − cmin )

(13)

where cˆ is the normalised vector, cmin and cmax are the minimum and maximum values of the vector c, respectively. Similarly, elements of w, the weight vector, have been normalised to sum to unity as wi =

wi S3 wi

(14)

The limit in (14) extends up to 3, because the weight vector w contains three elements in this work, corresponding to each criterion. In Fig. 3, the flow diagram of the proposed multi-criteria multi-stage optimal PMU placement method is shown. The dotted rectangles in the flow diagram demarcate the use of ILP and MCDM model. As shown in Fig. 3, ILP is used to solve the optimal PMU placement problem to ensure complete system observability and MCDM is used to rank the optimal PMU locations, taking the three proposed IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181– 190 doi: 10.1049/iet-gtd.2009.0709

indices, viz. BOI, VOI and TOI into consideration. Once the overall ranking of the optimal PMU locations is calculated, n1 PMU locations with highest priority in the order of their relative ranking are considered in the first stage, then the next n2 PMUs with the next highest priority in the second stage, and so on.

4

Simulation results

The effectiveness of the proposed method for multi-stage optimal PMU placement has been studied on IEEE 14-bus, NE 39-bus [18] and NRPG [19] 246-bus Indian systems. The IEEE 14-bus system has five synchronous machines, three of which are synchronous condensers used for reactive power support. NE 39-bus system, having 10 generators, 19 loads and 36 transmission lines, represents a reduced model of the NE power system. NRPG is the biggest among all the five regional electricity boards in India. A single-line diagram of the system is shown in Fig. 4. The NRPG system comprises nine states and covers around 30% geographical area and 28% population of India. A reduced representation of the NRPG system has been considered, which consists of 246 buses (220 kV and 400 kV only) and 376 branches (lines/transformers). Details of the results obtained on the three test systems are discussed below. 4.1

Optimal PMU placement

The ILP is formulated for each test system using (1) – (3), as explained in Section 2. Further, zero-injection buses, which are analogous to the simple nodes, have a potential to reduce the number of PMUs required for complete system observability [12]. Therefore in the following zero-injection buses are considered for the three test systems. 185

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www.ietdl.org

Fig. 4 Single-line diagram of 246-bus NRPG system network [19] (400 kV and above)

† IEEE 14-bus system: {7}; † NE 39-bus system: {1, 9, 18}; † NRPG 246-bus system: {63, 75, 81, 102, 103, 104, 107, 122, 155, 180, 210, 226, 237, 241, 244}.

system. The aim of the following study is to rank these optimal PMU locations so that PMU installation can be carried out in stages. 4.2

The ILP is solved using CPLEX solver [14]. Table 1 shows the results of the optimal PMU placement. The results listed in Table 1 ensure complete system observability even under the outage of a single PMU or a transmission line in the 186 & The Institution of Engineering and Technology 2011

Multi-stage PMU installation

For the multi-stage optimal PMU installation, 7, 26 and 135 PMU locations, as obtained through the ILP (listed in Table 1), are considered for IEEE 14-bus, NE 39-bus and IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181 –190 doi: 10.1049/iet-gtd.2009.0709

www.ietdl.org Table 1

Optimal PMU locations

System

Number of PMUs with ILP

Optimal locations

7 26 135

2, 4, 5, 6, 9, 11, 13 2, 3, 6, 8, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 2, 4, 5, 6, 7, 10, 11, 12, 15, 21, 23, 24, 27, 30, 31, 33, 34, 38, 40, 41, 44, 45, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 60, 61, 62, 65, 69, 71, 73, 74, 77, 80, 84, 87, 88, 89, 91, 92, 93, 95, 96, 98, 101, 105, 106, 108, 109, 111, 113, 117, 118, 119, 120, 123, 124, 125, 126, 128, 129, 132, 133, 134, 135, 139, 140, 141, 142, 144, 145, 147, 148, 149, 151, 153, 156, 157, 158, 159, 160, 163, 167, 168, 169, 171, 172, 173, 174, 176, 177, 178, 181, 185, 187, 188, 189, 190, 191, 193, 194, 195, 197, 199, 201, 202, 203, 206, 207, 208, 213, 216, 217, 218, 219, 223, 224, 225, 228, 229, 234, 235, 236, 238, 242, 243, 246

IEEE 14-bus NE 39-bus NRPG 246-bus

NRPG 246-bus systems, respectively. The PMU installation is assumed to be carried out as per the following scheme: † 14-Bus system: in total two stages, with four PMUs in the first stage and three PMUs in the second stage. † 39-Bus system: in total four stages, with seven PMUs in the first three stages and five PMUs in the last stage. † 246-Bus system: in total five stages with 27 PMUs in each stage.

Further, ranking of each PMU location is calculated under two different cases: (i) using unequal weightage (UW) to each criterion calculated using the method described in Section 3.2 and (ii) with equal weightage (EW). For case (i), a pairwise matrix is formed, as shown in Table 4. In this case, the tie-line observability criterion is considered to be an extremely important factor as compared

The proposed scheme is first demonstrated on IEEE 14-bus system. The three criteria, as proposed in Section 3, are calculated for all the seven optimal PMU buses listed in Table 1. The normalised BOI, calculated for each PMU bus is listed in Table 2. The electrical distance between any two nodes is calculated as explained in Section 3.1 and the values are listed in Table 3. Further, VOI is calculated at each PMU bus using (9), and the normalised values are shown in Table 2. These criteria values are normalised in the range of 0 – 1 using (13). In order to calculate the TOI, IEEE 14-bus system is decomposed into two sub-networks using an ILP eigenvector-based approach [20]. The partitioned network is shown in Fig. 5a. There are four tie-lines in the partitioned network, viz. 2 – 3, 4 – 5, 10– 11 and 13– 14. TOI, as calculated at each PMU bus, is indicated in Fig. 5a. Table 2

Normalised selection criteria and PLPI values for IEEE 14-bus system for UW and EW PMU bus

BOI

VOI

TOI

PLPIUW

PLPIEW

2 4 5 6 9 11 13

0.667 1.0 0.667 0.667 0.667 0 0.333

1.0 0.1891 0 0.2991 0.2236 0.6495 0.5093

1.0 1.0 0.5 0 0.5 1.0 1.0

0.9789 0.7847 0.3777 0.1213 0.4371 0.8440 0.8277

0.8889 0.7297 0.3889 0.3219 0.4634 0.5498 0.6142

Table 3 PMU bus 2 4 5 6 9 11 13

Fig. 5 MAUT results in IEEE 14-bus system a Calculation of tie-line observability index in IEEE 14-bus system b Overall PLPI value at different optimal PMU locations in IEEE 14-bus system for weighted and unweighted case

Normalised electrical distance in IEEE 14-bus 2

4

5

6

9

11

13

0 0.43615 0.36617 0.73893 0.75744 0.86686 0.85774

0.43615 0 0.14801 0.45291 0.38129 0.54247 0.5583

0.36617 0.14801 0 0.39474 0.44557 0.53597 0.51813

0.73893 0.45291 0.39474 0 0.39145 0.26702 0.16481

0.75744 0.38129 0.44557 0.39145 0 0.3293 0.44059

0.86686 0.54247 0.53597 0.26702 0.3293 0 0.392

0.85774 0.5583 0.51813 0.16481 0.44059 0.392 0

IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181– 190 doi: 10.1049/iet-gtd.2009.0709

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www.ietdl.org Table 4

Pairwise matrix for weight calculation with preference given to tie-line observability BOI

VOI

TOI

1 5 9

1/5 1 3

1/9 1/3 1

BOI VOI TOI

Table 5

Multi-stage optimal PMU placement in NE 39-bus system with UW {0.0629, 0.2654, 0.6716} and EW Ph-1

Ph-2

Ph-3

Fig. 6 Impact of multi-stage PMU placement on NE 39-bus system for unequal weightage case

Ph-4

UW

EW

UW

EW

UW

EW

UW

EW

17 8 6 16 10 29 2

17 8 16 6 2 26 29

25 26 12 13 34 20 37

10 25 19 13 20 23 12

19 38 31 36 23 22 32

22 34 3 14 37 38 31

35 3 14 30 33

36 32 35 30 33

(12) is solved and the PLPIUW of each PMU location is listed in Table 1. Similarly in case (ii), where all the criteria are given equal weightage, the utility values for each alternative are calculated and are shown as PLPIEW in Table 2. The PLPI values for the two cases are also compared in Fig. 5b. It is clear from the results that the first priority should be given (according to both the cases) to bus location-2 and the PMU must be placed at bus 2 in the first stage. The proposed methodology is also applied on NE 39-bus system. The test system is decomposed into three subnetworks using the ILP eigenvector-based multi-partitioning algorithm [20] and has eight tie-lines as 2 – 25, 9 – 39, 5 –6, 5 – 8, 17 – 18, 27– 28, 10– 11 and 12 – 13. Table 5 lists out results of the PMUs’ selection in four stages, as obtained by the MAUT calculations using (12) for the UW and EW cases.

to the bus connectivity. Similarly, tie-line observability was assumed to be moderately important as compared to the uniform distribution of PMUs. Accordingly, w3/w1 ¼ 9, w2/w1 ¼ 5 and w3/w2 ¼ 3. The pairwise matrix has maximum eigenvalue of 3.0291 and the corresponding right eigenvector is [0.0868, 0.3662, 0.9265]. Normalising to sum to unity, using (14), gives weights of each criterion as [0.0629, 0.2654, 0.6716]. With these values of the weights, Table 6

Ranking of optimal PMU locations in each stage in NRPG 246-bus system with UW {0.0629, 0.2654, 0.6716} and EW

Ph-1

Ph-2

Ph-3

Ph-4

Ph-5

UW

EW

UW

EW

UW

EW

UW

EW

UW

EW

34 65 80 245 48 235 229 118 69 10 242 133 61 158 56 109 44 71 132 62 174 173 178 171 168 194 160

65 34 80 229 235 245 48 118 194 56 242 160 69 109 157 213 84 141 133 10 21 168 158 199 191 187 54

177 188 156 199 172 157 163 213 84 191 141 201 187 54 21 238 169 167 113 129 190 181 91 74 88 140 139

238 44 113 129 190 91 74 88 71 140 132 139 27 40 201 181 195 61 197 185 216 174 234 217 125 203 62

224 195 197 246 108 27 40 120 185 216 234 207 159 217 105 126 125 202 203 106 189 193 225 219 55 123 7

219 7 11 15 24 33 173 171 169 167 207 105 126 202 177 106 193 188 55 101 57 98 96 89 73 223 163

11 15 24 33 101 57 206 98 96 89 111 45 50 176 73 223 60 128 51 236 147 92 208 95 218 53 134

147 134 178 108 225 156 206 172 111 45 50 176 60 128 236 92 95 218 87 93 228 144 142 145 117 119 151

124 47 87 93 228 144 142 145 117 119 151 52 148 77 6 23 149 153 135 2 4 5 12 30 31 38 41

148 224 6 23 246 120 159 189 123 51 208 53 124 47 52 77 149 153 135 2 4 5 12 30 31 38 41

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Fig. 7 Impact of multi-stage PMU placement on NRPG 246-bus system for unequal weightage case

Fig. 6 shows the multi-stage PMU placement performance results. Two bars in Fig. 6 correspond to the total number of tie-line buses observed and total BOI achieved in the respective stage. It can be noticed from Table 1 that out of 16 tie-line buses, only eight are in the list of optimal PMU locations. Rest of the eight tie-line buses will either be observed by the PMU placed on the other end of the tieline or by the PMUs placed at the neighbouring buses. From the MAUT results, it is observed that out of the 16 tie-line buses, 12 tie-line buses are observed during the first stage and four tie-line buses are observed during the second stage of the PMU placement, as shown in Fig. 6. Similarly, PMU placed in the first stage results in a total BOI value of 25. The total BOI value in subsequent stages keeps on increasing and it has been observed that by the end of the third stage of PMU placement, all the buses are observed. The proposed method is also tested on the practical NRPG 246-bus Indian system. The system is decomposed into nine subsystems, representing separate state electricity networks, and the total number of tie-lines obtained is 28. The total number of PMUs to be placed in the system is 135, as obtained in Section 4.1. The PMU placement is carried out in five stages. The multi-stage PMU placement results for the NRPG system, as obtained through (12), are listed in Table 6 for both the UW and EW cases. Similar to Fig. 6, Fig. 7 depicts the total number of tie-line buses observed and the total BOI value in each stage of PMU placement in the NRPG 246-bus system. It can be seen from Fig. 7 that all the tie-lines are observed by the end of the first stage of the PMU placement itself. The simulation results on the three test systems reveal that the complete observability is achieved much before the last stage of installation of PMUs and similarly, all the tie-lines get observed well before the last stage of the PMU placement. Since the PMU placement is robust against a PMU and a single line outage, the total number of optimal PMU locations is approximately 50% of the number of buses in the system and redundant PMUs are placed towards the last stages. It is also to be noted that there might be other critical factors also like monitoring of the critical corridors, critical oscillatory modes and so on, which the utilities might like to consider while placing the PMUs. These factors can easily be incorporated in the proposed MCDM framework, and the priority of these criteria can, further, be raised (if needed) by assigning higher weights to them.

5

Conclusions

In the proposed work, a multi-stage PMU placement scheme is developed, which combines ILP and MCDM approach. IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181– 190 doi: 10.1049/iet-gtd.2009.0709

The ILP determines a set of optimal PMU locations to make the system completely observable even under a branch outage or a PMU failure. The MCDM employs utility theory to find weights to the three criteria considered for prioritising the optimal PMU locations. The criteria, used in the present work, include tie-line oscillations observability, voltage control area observability and bus voltage observability. Based on these criteria, three indices are proposed, viz. BOI, VOI and TOI. Finally, the proposed indices are used in the MCDM process for ranking the optimal PMU locations and, thereby, selecting the PMU locations in each stage. The proposed scheme is demonstrated on three systems. The simulation results show that the proposed method can benefit utilities in deciding multi-stage PMU installations as it facilitates in gaining a maximum advantage from a PMU installation in terms of the bus voltage, voltage control area and tie-line observability. At the end of multi-stage PMU placement, the proposed PMU placement technique will render complete system observability with only phasor measurements.

6

References

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IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 2, pp. 181 –190 doi: 10.1049/iet-gtd.2009.0709